Skyscrapers Are Standard: An Overview

Abstract

‘Skyscrapers Are Standard: An Overview’ provides a detailed outline of the upcoming proof that a skyscraper is homeomorphic to the standard 2-handle, relative to the attaching region. Since the proof is technically challenging, this chapter serves to introduce the key ideas without cumbersome notation. The key points to keep in mind during the proof are enumerated. Briefly, the proof consists of finding a common subset, called the design, of both the given skyscraper and the standard 2-handle. This is accomplished by studying the vertical boundaries of skyscrapers. Next, one would wish to show that the decompositions of the skyscraper and the 2-handle, arising from the connected components of the complements of the common subset, shrink. The decompositions have to be modified to achieve this.